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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 21st 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexI added a definition-section to [[formal scheme]] with the four equivalent definitions of a k-formal scheme from [Demazure, lectures on p-divisible groups](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups). There is some overlap with the section on Noetherian formal schemes now.

   I added a definition-section to formal scheme with the four equivalent definitions of a k-formal scheme from Demazure, lectures on p-divisible groups. There is some overlap with the section on Noetherian formal schemes now.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 21st 2012
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   Author: zskoda
   Format: MarkdownItexThank you. Nice to have the details of the classical case in the entry. I think that the assertion that it is a ringed space is a bit imprecise. The classical cases may be realized as a topological space with a sheaf of __topological rings__, so not just rings. The fact that in the EGA generality the topological ring has a rather simple I-adic topology does not matter: the I-adic filtration is still part of the definition. In greater generality formal schemes are usually defined as some nice subcategory of ind-schemes.

   Thank you. Nice to have the details of the classical case in the entry.

   I think that the assertion that it is a ringed space is a bit imprecise. The classical cases may be realized as a topological space with a sheaf of topological rings, so not just rings. The fact that in the EGA generality the topological ring has a rather simple I-adic topology does not matter: the I-adic filtration is still part of the definition.

   In greater generality formal schemes are usually defined as some nice subcategory of ind-schemes.

3. • CommentRowNumber3.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 21st 2012
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   Author: hilbertthm90
   Format: MarkdownItexInteresting, apparently when I made [[p-divisible group]] I never used the term formal scheme to actually link to this page. I do reference Demazure, which I didn't realize was online.

   Interesting, apparently when I made p-divisible group I never used the term formal scheme to actually link to this page. I do reference Demazure, which I didn’t realize was online.

4. • CommentRowNumber4.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 22nd 2012
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   Author: Stephan A Spahn
   Format: MarkdownItex> I think that the assertion that it is a ringed space is a bit imprecise... Maybe we could write it is a ''ring-objected space'' or the like. > In greater generality formal schemes are usually defined as some nice subcategory of ind-schemes. In maybe even greater generality it is a formally smooth object in a cohesive topos; if I remember this is an example in Rosenberg-Kontsevich, noncommutative spaces.

   I think that the assertion that it is a ringed space is a bit imprecise…

   Maybe we could write it is a ”ring-objected space” or the like.

   In greater generality formal schemes are usually defined as some nice subcategory of ind-schemes.

   In maybe even greater generality it is a formally smooth object in a cohesive topos; if I remember this is an example in Rosenberg-Kontsevich, noncommutative spaces.

5. • CommentRowNumber5.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 22nd 2012
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   Author: Stephan A Spahn
   Format: MarkdownItex> Interesting, apparently when I made [[p-divisible group]] I never used the term formal scheme to actually link to this page. You did not indicated this, but is there a generalization of $p$-divisibility to other contexts than formal schemes? I guess in principle one can define this by taking an endomorphism with comparable properties instead of the Frobenius morphism.

   Interesting, apparently when I made p-divisible group I never used the term formal scheme to actually link to this page.

   You did not indicated this, but is there a generalization of $p$-divisibility to other contexts than formal schemes? I guess in principle one can define this by taking an endomorphism with comparable properties instead of the Frobenius morphism.